On the characterization of Triebel--Lizorkin type spaces of analytic functions

TL;DR

This paper explores various characterizations of Triebel--Lizorkin spaces of analytic functions on the unit disc, extending known factorization results to vector-valued functions and clarifying conditions for their validity.

Contribution

It generalizes a key factorization theorem for Triebel--Lizorkin spaces to vector-valued functions and identifies conditions under which the factorization holds or fails.

Findings

Factorization theorem extends to Banach space-valued functions with correct vector placement.

Factorization fails for vector-valued functions when the vector-valuedness is in the wrong factor.

Provides detailed descriptions of Triebel--Lizorkin spaces, including for generalized scales like Q-spaces.

Abstract

We consider different characterizations of Triebel--Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called $Q$-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel--Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness "sits in the right factor". On the other hand, if we impose vector-valuedness on the "wrong" factor, then the factorization fails even for separable Hilbert spaces.